2,901 research outputs found
Exact Real Arithmetic with Perturbation Analysis and Proof of Correctness
In this article, we consider a simple representation for real numbers and
propose top-down procedures to approximate various algebraic and transcendental
operations with arbitrary precision. Detailed algorithms and proofs are
provided to guarantee the correctness of the approximations. Moreover, we
develop and apply a perturbation analysis method to show that our approximation
procedures only recompute expressions when unavoidable.
In the last decade, various theories have been developed and implemented to
realize real computations with arbitrary precision. Proof of correctness for
existing approaches typically consider basic algebraic operations, whereas
detailed arguments about transcendental operations are not available. Another
important observation is that in each approach some expressions might require
iterative computations to guarantee the desired precision. However, no formal
reasoning is provided to prove that such iterative calculations are essential
in the approximation procedures. In our approximations of real functions, we
explicitly relate the precision of the inputs to the guaranteed precision of
the output, provide full proofs and a precise analysis of the necessity of
iterations
Delta-Complete Decision Procedures for Satisfiability over the Reals
We introduce the notion of "\delta-complete decision procedures" for solving
SMT problems over the real numbers, with the aim of handling a wide range of
nonlinear functions including transcendental functions and solutions of
Lipschitz-continuous ODEs. Given an SMT problem \varphi and a positive rational
number \delta, a \delta-complete decision procedure determines either that
\varphi is unsatisfiable, or that the "\delta-weakening" of \varphi is
satisfiable. Here, the \delta-weakening of \varphi is a variant of \varphi that
allows \delta-bounded numerical perturbations on \varphi. We prove the
existence of \delta-complete decision procedures for bounded SMT over reals
with functions mentioned above. For functions in Type 2 complexity class C,
under mild assumptions, the bounded \delta-SMT problem is in NP^C.
\delta-Complete decision procedures can exploit scalable numerical methods for
handling nonlinearity, and we propose to use this notion as an ideal
requirement for numerically-driven decision procedures. As a concrete example,
we formally analyze the DPLL framework, which integrates Interval
Constraint Propagation (ICP) in DPLL(T), and establish necessary and sufficient
conditions for its \delta-completeness. We discuss practical applications of
\delta-complete decision procedures for correctness-critical applications
including formal verification and theorem proving.Comment: A shorter version appears in IJCAR 201
Efficient algorithms for computing the Euler-Poincar\'e characteristic of symmetric semi-algebraic sets
Let be a real closed field and
an ordered domain. We consider the algorithmic problem of computing the
generalized Euler-Poincar\'e characteristic of real algebraic as well as
semi-algebraic subsets of , which are defined by symmetric
polynomials with coefficients in . We give algorithms for computing
the generalized Euler-Poincar\'e characteristic of such sets, whose
complexities measured by the number the number of arithmetic operations in
, are polynomially bounded in terms of and the number of
polynomials in the input, assuming that the degrees of the input polynomials
are bounded by a constant. This is in contrast to the best complexity of the
known algorithms for the same problems in the non-symmetric situation, which
are singly exponential. This singly exponential complexity for the latter
problem is unlikely to be improved because of hardness result
(-hardness) coming from discrete complexity theory.Comment: 29 pages, 1 Figure. arXiv admin note: substantial text overlap with
arXiv:1312.658
Complete Subdivision Algorithms, II: Isotopic Meshing of Singular Algebraic Curves
Given a real valued function f(X,Y), a box region B_0 in R^2 and a positive
epsilon, we want to compute an epsilon-isotopic polygonal approximation to the
restriction of the curve S=f^{-1}(0)={p in R^2: f(p)=0} to B_0. We focus on
subdivision algorithms because of their adaptive complexity and ease of
implementation. Plantinga and Vegter gave a numerical subdivision algorithm
that is exact when the curve S is bounded and non-singular. They used a
computational model that relied only on function evaluation and interval
arithmetic. We generalize their algorithm to any bounded (but possibly
non-simply connected) region that does not contain singularities of S. With
this generalization as a subroutine, we provide a method to detect isolated
algebraic singularities and their branching degree. This appears to be the
first complete purely numerical method to compute isotopic approximations of
algebraic curves with isolated singularities
Computing the Lambert W function in arbitrary-precision complex interval arithmetic
We describe an algorithm to evaluate all the complex branches of the Lambert
W function with rigorous error bounds in interval arithmetic, which has been
implemented in the Arb library. The classic 1996 paper on the Lambert W
function by Corless et al. provides a thorough but partly heuristic numerical
analysis which needs to be complemented with some explicit inequalities and
practical observations about managing precision and branch cuts.Comment: 16 pages, 4 figure
A probabilistic approach to reducing the algebraic complexity of computing Delaunay triangulations
Computing Delaunay triangulations in involves evaluating the
so-called in\_sphere predicate that determines if a point lies inside, on
or outside the sphere circumscribing points . This
predicate reduces to evaluating the sign of a multivariate polynomial of degree
in the coordinates of the points . Despite
much progress on exact geometric computing, the fact that the degree of the
polynomial increases with makes the evaluation of the sign of such a
polynomial problematic except in very low dimensions. In this paper, we propose
a new approach that is based on the witness complex, a weak form of the
Delaunay complex introduced by Carlsson and de Silva. The witness complex
is defined from two sets and in some metric space
: a finite set of points on which the complex is built, and a set of
witnesses that serves as an approximation of . A fundamental result of de
Silva states that if .
In this paper, we give conditions on that ensure that the witness complex
and the Delaunay triangulation coincide when is a finite set, and we
introduce a new perturbation scheme to compute a perturbed set close to
such that . Our perturbation
algorithm is a geometric application of the Moser-Tardos constructive proof of
the Lov\'asz local lemma. The only numerical operations we use are (squared)
distance comparisons (i.e., predicates of degree 2). The time-complexity of the
algorithm is sublinear in . Interestingly, although the algorithm does not
compute any measure of simplex quality, a lower bound on the thickness of the
output simplices can be guaranteed.Comment: 24 page
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